{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T16:54:27Z","timestamp":1771001667125,"version":"3.50.1"},"reference-count":26,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2005,7,8]],"date-time":"2005-07-08T00:00:00Z","timestamp":1120780800000},"content-version":"vor","delay-in-days":4025,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[1994,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Recently, Freund and Nachtigal proposed the quasi\u2010minimal residual algorithm (QMR) for solving general nonsingular non\u2010Hermitian linear systems. The method is based on the Lanczos process, and thus it involves matrix\u2014vector products with both the coefficient matrix of the linear system and its transpose. Freund developed a variant of QMR, the transpose\u2010free QMR algorithm (TFQMR), that only requires products with the coefficient matrix. In this paper, the use of QMR and TFQMR for solving singular systems is explored. First, a convergence result for the general class of Krylov\u2010subspace methods applied to singular systems is presented. Then, it is shown that QMR and TFQMR both converge for consistent singular linear systems with coefficient matrices of index 1. Singular systems of this type arise in Markov chain modeling. For this particular application, numerical experiments are reported.<\/jats:p>","DOI":"10.1002\/nla.1680010406","type":"journal-article","created":{"date-parts":[[2005,11,1]],"date-time":"2005-11-01T17:51:00Z","timestamp":1130867460000},"page":"403-420","source":"Crossref","is-referenced-by-count":52,"title":["On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling"],"prefix":"10.1002","volume":"1","author":[{"given":"Roland W.","family":"Freund","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marlis","family":"Hochbruck","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2005,7,8]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01934996"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1080\/15326348908807115"},{"key":"e_1_2_1_4_2","unstructured":"V. A.BarkerandB. F.Nielsen.The analysis of a manufacturing unit by sparse matrix techniques. In Proceedings of the conference \u2018Numerical Solution of Markov Chains\u2019 Raleigh NC January1990."},{"key":"e_1_2_1_5_2","volume-title":"Nonnegative Matrices in the Mathematical Sciences","author":"Berman A.","year":"1979"},{"key":"e_1_2_1_6_2","volume-title":"Generalized Inverses of Linear Transformations","author":"Campbell S. L.","year":"1979"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01404464"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1137\/0914029"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8619-2_5"},{"key":"e_1_2_1_10_2","volume-title":"Krylov\u2010subspace methods for non\u2010Hermitian p\u2010cyclic matrices. Numerical Analysis Manuscript","author":"Freund R. 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Ph.D. dissertation Universit\u00e4t Karlsruhe 1989."},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511810817"},{"key":"e_1_2_1_18_2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511840371"},{"key":"e_1_2_1_19_2","doi-asserted-by":"publisher","DOI":"10.6028\/jres.045.026"},{"key":"e_1_2_1_20_2","doi-asserted-by":"publisher","DOI":"10.2307\/2007796"},{"key":"e_1_2_1_21_2","doi-asserted-by":"publisher","DOI":"10.1287\/opre.40.6.1156"},{"key":"e_1_2_1_22_2","volume-title":"Discrete Markov Chains","author":"Romanovsky V. I.","year":"1970"},{"key":"e_1_2_1_23_2","unstructured":"Y.Saad.SPARSKIT: a basic tool kit for sparse matrix computations. Technical Report 90.20 RIACS NASA Ames Research Center 1990."},{"key":"e_1_2_1_24_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90169-X"},{"key":"e_1_2_1_25_2","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"e_1_2_1_26_2","volume-title":"Non\u2010Negative Matrices","author":"Seneta E.","year":"1973"},{"key":"e_1_2_1_27_2","doi-asserted-by":"publisher","DOI":"10.1137\/0910004"}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.1680010406","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.1680010406","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T01:09:57Z","timestamp":1698109797000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.1680010406"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,7]]},"references-count":26,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1994,7]]}},"alternative-id":["10.1002\/nla.1680010406"],"URL":"https:\/\/doi.org\/10.1002\/nla.1680010406","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"value":"1070-5325","type":"print"},{"value":"1099-1506","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,7]]}}}